We are addressing the key questions of:
| Median | Upper 50 CI | Lower 50 CI | |
|---|---|---|---|
| Peak Hospitalizations | 6,118.00 | 9,796.00 | 2,554.00 |
| Deaths by August 1, 2020 | 21,863.00 | 31,521.00 | 12,765.50 |
| Detected Illnesses by August 1,2020 | 624,165.00 | 901,032.75 | 364,527.00 |
| Total Illnesses by August 1, 2020 | 5,413,184.00 | 6,716,862.75 | 3,208,621.25 |
| Proportion of Cases Detected (%) | 13.22 | 16.04 | 10.45 |
| CFR Based on Observed Illnesses (%) | 3.87 | 4.77 | 2.78 |
| CFR Based on Total Illnesses (%) | 0.48 | 0.61 | 0.35 |
| R0 - before social distancing | 3.38 | 3.79 | 2.96 |
| % Reduction in Social Contacts (March 15 - ) | 59.82 | 54.26 | 63.59 |
Dashed line = Maximum possible capacity (i.e., total licensed hospital beds, ICU beds, ventilators) in L.A. County
Demonstrating model fit against COVID-19 data for Los Angeles, for the following variables:
COVID-19 data is shown as black dots in the figures below.
We analyze how population prevalence of known COVID-19 risk factors: advanced age, existence of other health conditions or comorbidities, smoking status, and obesity status, affect COVID-19 illness trajectories in L.A. County and spatial subdivisions.
First, we estimate the conditional probability of COVID illness severity given combinations of risk factors. We categorize the population into a number of risk profiles, representing different combinations of known COVID-19 risk factors: advanced age, existence of other health conditions or comorbidities, smoking status, and obesity status. Using previous COVID-19 studies reporting the marginal risk of severe COVID-19 outcomes given individual risk factors, we develop a statistical model to estimate the probability of COVID illness trajectories for individuals having or not having combinations of risk factors represented by these risk profiles. Specifically, we estimate the probability that individuals within a specific risk group are admitted to hospital given having acquired illness \(Pr(Hospital | Illness, Profile_i)\), are admitted to the ICU given admittance to hospitalized \(Pr(ICU | Hospital, Profile_i)\), and that die given being admitted to the ICU \(Pr(Death | ICU, Profile_i)\). More information is provided below under Methods and Data.
For the analysis below, we have grouped the multiple risk profiles into 5 key risk groups according to similar within-group levels of the probabilities \(Pr(Hospital | Illness, Profile_i)\), \(Pr(ICU | Hospital, Profile_i)\), and \(Pr(Death | ICU, Profile_i)\).
Second, we use these probabilities to estimate the proportion of each risk group that will make up the resulting cohorts of COVID patients admitted to hospital, admitted to ICU, or that die within the L.A. County population, based on the prevalence of each risk group in the population.
Results are also presented for each Service Planning Area (SPA) population within L.A. County. A SPA is a specific geographical region within Los Angeles County used by the Department of Public Health to plan and provide health services. There are 8 SPAs in Los Angeles County.
Here we summarize our estimated parameter values for key epidemic and model quantities:
Because our model is stochastic and we are using Bayesian techniques for parameter estimation, each posterior parameter estimate is represented by a distribution of likely values.
This table summarizes key statistics of each estimated parameter: the mean and the standard deviation (sd).
| R0 | Prop. cases detected (r) | Frac R0 Mar11 | Pr(Death|ICU) | Pr(Hospital|Illness) | Pr(ICU|Hospital) | Pr(Ventilation|ICU) | Frac R0 Apr23 | |
|---|---|---|---|---|---|---|---|---|
| mean | 3.36 | 0.14 | 0.42 | 0.54 | 0.21 | 0.36 | 0.74 | 0.40 |
| sd | 0.61 | 0.05 | 0.06 | 0.16 | 0.03 | 0.04 | 0.08 | 0.06 |
Information informing prior distribution - \(R0\) prior estimate is based on values for \(R0\) estimated from other published studies on COVID-19.
Information informing this parameter’s prior distribution:
We use mobility data to narrow the specification of the reduction in the average number of new infections due to an infected person (R0) in a completely susceptible population under recent social distancing restrictions.
Effectively, reductions in mobility correspond to a proportional reduction in R0
Reduction in mobility observed in LA County: Source: Assessing changes in commuting and individual mobility in major metropolitan areas in the United States during the COVID-19 outbreak
Our modeled reduction in R0 timeline:
We use previous studies to narrow the specification of the probability of hospitalization given illness, admittance to the intensive care unit (ICU) given being in hospital, ventilation given being in ICU, and death given being in ICU by incorporating risk factors, including age, sex, smoking and other comorbidities. The prevalence of these risk factors in Los Angeles County is also included.
Studies on COVID-19 clinical presentation and trajectories to inform the probability of hospitalization, ICU, and ventilation based on single risk factors: - Guan, Wei-jie, et al. “Clinical characteristics of coronavirus disease 2019 in China.” New England Journal of Medicine (2020). - Petrilli, Christopher M., et al. “Factors associated with hospitalization and critical illness among 4,103 patients with COVID-19 disease in New York City.” medRxiv (2020).
Prevalence data sources: - Los Angeles County Health Survey - UCLA California Health Information Survey
What a stochastic model allows:
Compartmental model flow diagram
\[ \begin{align*} dS/dt &= -\beta S(I+A)\\ dE/dt &= \beta S(I+A) - \tfrac{1}{d_{EI}}E\\ dA/dt &= \tfrac{1-r}{d_{EI}}E - \tfrac{1}{d_{IR}}A\\ dI/dt &= \tfrac{r}{d_{EI}}E - (\tfrac{\alpha}{d_{IH}}\tfrac{1-\alpha}{d_{IR}})I\\ dH/dt &= \alpha (\tfrac{\alpha}{d_{IH}}\tfrac{1-\alpha}{d_{IR}})I - (\tfrac{\kappa}{d_{HQ}}\tfrac{1-\kappa}{d_{HR}})H \\ dQ/dt &= \kappa (\tfrac{\kappa}{d_{HQ}}\tfrac{1-\kappa}{d_{HR}})H - (\tfrac{\delta}{d_{QD}}\tfrac{1-\delta}{d_{QR}})Q \\ dV/dt &= p_V Q\\ dD/dt &= \delta (\tfrac{\delta}{d_{QD}}\tfrac{1-\delta}{d_{QR}})Q\\ dR/dt &= (1-\alpha) (\tfrac{\alpha}{d_{IH}}\tfrac{1-\alpha}{d_{IR}})I + (1-\kappa) (\tfrac{\kappa}{d_{HQ}}\tfrac{1-\kappa}{d_{HR}})H + (1-\delta)(\tfrac{\delta}{d_{QD}}\tfrac{1-\delta}{d_{QR}})Q + \tfrac{1}{d_{IR}}A \ \end{align*} \]
\[ R0 = \beta ({\frac{r}{\tfrac{\alpha}{d_{IH}}+\tfrac{1-\alpha}{d_{IR}}}+ (1-r){d_{IR}}}) \\ N=S+E+A+I+H+Q+D+R \]
| Parameter | Description | Value |
|---|---|---|
| \(R0\) | Basic reproductive number | Estimated |
| \(\beta\) | transmission rate | Analytically derived from model and R0 |
| \(d_{EI}\) | days between exposure and infectivity (incubation period) | 5 days |
| \(d_{IH}\) | days between symptom onset and hospitalization (if required) | 10 days |
| \(d_{IR}\) | days between symptom onset and recovery (if not hospitalized) | 7 days |
| \(d_{HQ}\) | days between hospitalization and ICU (if required) | 1 days |
| \(d_{QR}\) | days between hospitalization and recovery (if ICU not required) | 12 days |
| \(d_{QD}\) | days between ICU and fatality | 8 days |
| \(d_{QR}\) | days between ICU and recovery | 7 days |
| \(\alpha\) | probability infected (I) requires hospitalization (vs. recovers) | Estimated |
| \(\kappa\) | probability hospitalized (H) requires ICU (vs. recovers) | Estimated |
| \(\delta\) | probability ICU (Q) patient dies | Estimated |
| \(p_V\) | probability ventilation (V) required given ICU | Estimated |
| \(N\) | Total population size | |
| \(S\) | Susceptible population | |
| \(E\) | Exposed not yet infectious | |
| \(A\) | Infected, unobserved | |
| \(I\) | Infected, observed | |
| \(H\) | In Hospital | |
| \(Q\) | In ICU | |
| \(V\) | On ventilator | |
| \(D\) | Dead | |
| \(R\) | Recovered/removed |
Model parameters - fixed, taken from literature: - Transition times between compartments - Sources provided at this link
Model Parameters — estimated by our model - \(R0\), the reproductive number or average number of new infections generated by an infected person in a completely susceptible population - \(r\), the proportion of illnesses that are observed - \(Frac_{R0}\), the reduction in the initial R0 due to social distancing - \(\alpha\), the probability of hospitalization given illness, i.e. \(Pr(Hospital | Illness)\) - \(\kappa\), the probability of ICU care necessary given hospitalization, i.e. \(Pr(ICU | Hospital)\) - \(p_v\), the probability of ventilation given ICU care, i.e. \(Pr(Ventilation | ICU)\) - \(\delta\), the probability of death given ICU care, i.e. \(Pr(Death | ICU)\)
We analyze how population prevalence of known COVID-19 risk factors: advanced age, existence of other health conditions or comorbidities, smoking status, and obesity status, affect COVID-19 illness trajectories in L.A. County and spatial subdivisions.
(1) Estimating the conditional probability of COVID illness severity given combinations of risk factors
\[ \begin{align*} Pr(Hospital | Illness) = \sum_i Pr(Group_i | Illness)Pr(Hospital|Group_i,Illness) \end{align*} \] We assume that the prevalence of the risk group in the ill population, \(Pr(Group_i|Illness)\), is equal to the prevalence of the group in the general population of L.A. County, i.e. \(Pr(Group_i)\). We again borrow the correlation structure between risk factors derived from the NHANES cohort to estimate the population prevalence \(Pr(Group_i)\) from available data on the prevalence of individual risk factors.
The same approach is applied to estimate \(Pr(ICU | Hospital)\) and \(Pr(Death|ICU)\): \[ \begin{align*} Pr(ICU | Hosptial) = \sum_i Pr(Group_i | Hospital)Pr(ICU|Group_i,Hospital)\\ Pr(Death | ICU) = \sum_i Pr(Group_i | ICU)Pr(Death|Group_i,ICU) \end{align*} \]
(2) Estimating the proportion of each risk group that will make up the cohorts of COVID patients admitted to hospital, admitted to ICU, or that die in L.A. County and SPAs
Data sources
Specification of stochastic model
## ```r
##
## # TRANSITION EQUATIONS
##
## ## Core equations for transitions between compartments:
## update(S) <- S - n_SE
## update(E) <- E + n_SE - n_Eout
## update(I) <- I + n_EoutI - n_Iout
## update(A) <- A + n_EoutA - n_AR
## update(H) <- H + n_IoutH - n_Hout
## update(Q) <- Q + n_HoutQ - n_Qout
## update(D) <- D + n_QoutD
## update(R) <- R + n_IoutR + n_HoutR + n_QoutR + n_AR
##
## ## Htot = H + Q
## update(Htot) <- H + Q + n_IoutH - n_HoutR - n_Qout # Htot represents all in Hospital: Non-ICU + ICU
##
## ## Ventilators (tracking as frac of Q, do not go to other compartments)
## update(V) <- p_QV*Q #V + n_QV - n_Vout
##
## ## Tracking cumulative numbers in compartments:
## update(Idetectcum) <- Idetectcum + n_EoutI
## update(Itotcum) <- Itotcum + n_Eout
## update(Htotcum) <- Htotcum + n_IoutH #Htotcum represents cumulative of all in Hospital: Non-ICU + ICU
## update(Qcum) <- Qcum + n_HoutQ
## update(Vcum) <- p_QV*Qcum #Vcum + n_QV
##
## ## New daily numbers
## output(I_detect_new) <- n_EoutI
## output(I_tot_new) <- n_Eout
## output(H_new) <- n_IoutH
## output(Q_new) <- n_HoutQ
## output(D_new) <- n_QoutD
## #output(d_EI_rand) <- d_EI
##
## ####################################################################################
##
## # PROBABILITIES
##
## ## Individual probabilities of transition:
## p_SE <- 1 - exp(-(Beta * (I+A)) / N) # S to E
## p_Eout <- 1 - exp(-1/d_EI) # E to I
## p_Iout <- 1 - exp(-((Alpha/d_IH) + ((1-Alpha)/d_IR))) #exp(-((1/d_IH) + (1/d_IR))) # I to H and R
## p_Hout <- 1 - exp(-((Kappa/d_HQ) + ((1-Kappa)/d_HR))) #exp(-((1/d_HQ) + (1/d_HR))) # H to Q and R
## p_Qout <- 1 - exp(-((Delta/d_QD) + ((1-Delta)/d_QR))) #exp(-((1/d_QD) + (1/d_QR))) # Q to D and R
## p_AR <- 1 - exp(-1/d_IR)
## #p_Vout <- 1 - exp(-1/d_V) # Leaving V
##
##
##
## # RANDOM DRAWS FOR NUMBERS CHANGING BETWEEN COMPARTMENTS
## ## Draws from binomial and multinomial distributions for numbers changing between compartments:
##
## ### S to E
## n_SE <- rbinom(S, p_SE)
##
## ### E to I and A
## n_Eout <- rbinom(E, p_Eout)
## n_EoutIA[] <- rmultinom(n_Eout, p_EoutIA)
## p_EoutIA[1] <- r
## p_EoutIA[2] <- 1-r
## dim(p_EoutIA) <- 2
## dim(n_EoutIA) <- 2
## n_EoutI <- n_EoutIA[1]
## n_EoutA <- n_EoutIA[2]
##
## ### A to R
## n_AR <- rbinom(A, p_AR)
##
## ### I to H and R
## n_Iout <- rbinom(I, p_Iout) # Total no. leaving I
## n_IoutHR[] <- rmultinom(n_Iout, p_IoutHR) # Divide total no. leaving I into I->H and I->R
## p_IoutHR[1] <- Alpha #(Alpha/d_IH)/((Alpha/d_IH) + ((1-Alpha)/d_IR)) # Goes to H and R with relative rates
## p_IoutHR[2] <- 1-Alpha #((1-Alpha)/d_IR)/((Alpha/d_IH) + ((1-Alpha)/d_IR)) # 1-p_IoutHR[1]
## dim(p_IoutHR) <- 2
## dim(n_IoutHR) <- 2
## n_IoutH <- n_IoutHR[1] # Total no. I->H
## n_IoutR <- n_IoutHR[2] # Total no. I->R
##
## ### H to Q and R
## n_Hout <- rbinom(H, p_Hout)
## n_HoutQR[] <- rmultinom(n_Hout, p_HoutQR)
## p_HoutQR[1] <- Kappa #(Kappa/d_HQ)/((Kappa/d_HQ) + ((1-Kappa)/d_HR))
## p_HoutQR[2] <- 1-Kappa #((1-Kappa)/d_HR)/((Kappa/d_HQ) + ((1-Kappa)/d_HR))
## dim(p_HoutQR) <- 2
## dim(n_HoutQR) <- 2
## n_HoutQ <- n_HoutQR[1]
## n_HoutR <- n_HoutQR[2]
##
## ### Q to D and R
## n_Qout <- rbinom(Q, p_Qout)
## n_QoutDR[] <- rmultinom(n_Qout, p_QoutDR)
## p_QoutDR[1] <- Delta #(Delta/d_QD)/((Delta/d_QD) + ((1-Delta)/d_QR))
## p_QoutDR[2] <- 1-Delta #((1-Delta)/d_QR)/((Delta/d_QD) + ((1-Delta)/d_QR))
## dim(p_QoutDR) <- 2
## dim(n_QoutDR) <- 2
## n_QoutD <- n_QoutDR[1]
## n_QoutR <- n_QoutDR[2]
##
## ### Q to V and Vout
## #n_QV <- rbinom(Q, p_QV)
## #n_Vout <- rbinom(V, p_Vout)
##
## ######################################################################
##
## # TOTAL POPULATION SIZE
## N <- S + E + I + A + H + Q + D + R
##
## ######################################################################
##
## # INITIAL STATES
## ## Core compartments
## initial(S) <- S_ini
## initial(E) <- E_ini
## initial(I) <- 0
## initial(A) <- 0
## initial(H) <- 0
## initial(Q) <- 0
## initial(D) <- 0
## initial(R) <- 0
## initial(V) <- 0
## initial(Htot) <- 0
##
## ## Cumulative counts
## initial(Idetectcum) <- 0
## initial(Itotcum) <- 0
## initial(Htotcum) <- 0
## initial(Qcum) <- 0
## initial(Vcum) <- 0
##
## ######################################################################
##
## # USER DEFINED PARAMETERS
## ## Default in parentheses:
##
## ### Initial conditions
## S_ini <- user(1e7) # susceptibles
## E_ini <- user(10) # infected
##
## ### Parameters - random
## #d_EI <- runif(3, 8)
##
## ### Parameters - fixed
## d_EI <- user(5.2) #days between exposure and infectivity (incubation period)
## d_IH <- user(10) #days between illness onset and hospitalization
## d_IR <- user(7) #days between illness onset and recovery (hospitalization not required)
## d_HQ <- user(1) #days between hospitalization start and ICU
## d_HR <- user(12) #days in hospital (ICU not required)
## d_QD <- user(8) #days in ICU before death (given death)
## d_QR <- user(7) #days in ICU before recovery (given recovery)
## #d_V <- user(3) #days on ventilator (within ICU)
##
## ### Parameters - weighted average risk probabilities: input from JAM + population prevalence
## Alpha <- user(0.14) #probability infected (I) requires hospitalization (vs. recovers)
## Kappa <- user(0.23) #probability hospitalized (H) requires ICU (vs. recovers)
## Delta <- user(0.06) #probability ICU (Q) patient dies
## p_QV <- user(0.667) #probability in ICU and requires ventilation
## r <- user(0.25)
##
## ### Other variables
## #R0 <- user(2.2) #Current estimates from other models
##
## ### Parameters - calculated from inputs
## #Br <- R0 * ( 1 / ( (r/ ((Alpha/d_IH) + ((1-Alpha)/d_IR))) + (1-r)*d_IR ))
##
##
##
## #########################################
## ### TIME VARYING BETA (INTERPOLATION) ###
## #########################################
##
## Beta <- interpolate(Beta_t, Beta_y,"linear")
##
## Beta_t[] <- user()# R0 * ((Alpha/d_IH)+((1-Alpha)/d_IR))
## Beta_y[] <- user()
## dim(Beta_t) <- user()
## dim(Beta_y) <- user()
##
##
##
## ```